Is there a whole number $x$ such that the sum of the digits of $x^2$ equals 44?
I would like someone to tell me if my thoughts are correct.
The remainder of a number a divided with 9 is the same as ...

Jack is looking at Anne, however Anne is looking at George. Jack is married, George isn't, and Anne's status is unknown. provided this info alone determine whether:
a) A married person looks at a non ...

What are some mathematical inequalities and theorems that follow using thermodynamics "proofs" (rigorous or just intuitive)?
Any suggested books on the matter?
For example, AM-GM inequality follows ...

Suppose there are three qualities of rice, A(1 dollar per Kg), b(2 dollar per Kg) and C(3 dollar per Kg). The salesmen want to mix these in a certain ratio a:b:c so as to make the price 2.5 dollar per ...

Chess has a limited number of maximum moves because of the 50-move rule (50 moves without any captures or pawn moves results in a draw). There are 30 capture-able pieces, and I've figured out that the ...

I'd like to think that I understand symmetry groups. I know what the elements of a symmetry group are - they are transformations that preserve an object or its relevant features - and I know what the ...

I've found a generalisation of Napoleon's theorem to general polygons.
Take any regular $n$-gon inscribed in a circle and stretch it (in any direction) so that the circle becomes an ellipse and the ...

Here is a puzzle that appeared in a Russian magazine named Kvantik (see Tanya Khovanova's Math Blog). [The trick lies in that we don't know exactly what the hedgehog knows at each stage. The symbology ...

You have $N$ wires that all extend from one location to a second distant location. The wire ends at both locations are unlabeled, and the goal is to label them all (on both ends) with distinct labels ...

Here's another problem, significantly harder than the first, but still accessible to target audience. The statement of the problem (i.e., northwest corner only) comes from a PennyDell puzzle magazine:
...

This is my first post. I hope it's acceptable.
EDIT Since there are people to whom such notation is foreign, I will point out that the problem represents KRRAEE / KMS, where PEI is the quotient and ...

The part four $K4$ of Sanborn sculpture, a sculpture located on the grounds of the CIA in Langley remains unsolved. As you can read in [1], Sanborn released a clue for the 64th-69th letters in part ...

I was looking at the solution for the Code Jam 2014 qualification question but the proof of correctness seems to be incomplete and I was wondering if anyone could help me with it. The full question ...

I was drawing some shapes during class, and I came across the following problem. If one takes steps of constant length, and one must deviate a constant angle $\alpha$ from one's previous step either ...

Occasionally I encounter a recurring instance of stupidity when I teach mathematics, namely that someone asks if we can play a game known in Danish under the name "bum", since that person thinks that ...

It is not too difficult to show that the Fibonacci numbers mod $b$ form a periodic sequence. I would like to say something interesting about the period.
There is a small shortcut to the brute-force ...

I'm looking for a formula for the following problem. Hopefully I can explain this clearly and it all makes sense. No, it's not my homework, it's part of a competition I'm involved with managing and ...

Here is the function:
$$f(a)=\sqrt{f(a)+\sqrt{f(f(a))+\sqrt{f(f(f(a)))+\cdots}}}$$
Is there another way to represent this function so that it only has $f(a)$ on one side and no $f(a)$'s on the other ...

If lines joining four corresponding vertices of two tetrahedrons are concurrent, then the lines of intersection of four corresponding planes are coplanar, and the converse also holds true.
What would ...

You can place weights on both side of weighing balance and you need to measure all weights between $1$ and $1000$. For example if you have weights $1$ and $3$, now you can measure $1,3$ and $4$ like ...

Consider the following game: Suppose the initial value of the pot is $ S $. Our player Josephine then rolls a fair $n$-sided die. If the roll is not $1$, then the pot is multiplied by that roll, and ...